Find the x and y Intercepts f(x)=6x^4+x^3-18x^2-3x

$f (x) = 6 x^{4} + x^{3} - 18 x^{2} - 3 x$

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 6 x^{4} + x^{3} - 18 x^{2} - 3 x$

Solve the equation.

Rewrite the equation as $6 x^{4} + x^{3} - 18 x^{2} - 3 x = 0$ .

$6 x^{4} + x^{3} - 18 x^{2} - 3 x = 0$

Factor the left side of the equation.

Factor $x$ out of $6 x^{4} + x^{3} - 18 x^{2} - 3 x$ .

Factor $x$ out of $6 x^{4}$ .

$x (6 x^{3}) + x^{3} - 18 x^{2} - 3 x = 0$

Factor $x$ out of $x^{3}$ .

$x (6 x^{3}) + x \cdot x^{2} - 18 x^{2} - 3 x = 0$

Factor $x$ out of $- 18 x^{2}$ .

$x (6 x^{3}) + x \cdot x^{2} + x (- 18 x) - 3 x = 0$

Factor $x$ out of $- 3 x$ .

$x (6 x^{3}) + x \cdot x^{2} + x (- 18 x) + x \cdot - 3 = 0$

Factor $x$ out of $x (6 x^{3}) + x \cdot x^{2}$ .

$x (6 x^{3} + x^{2}) + x (- 18 x) + x \cdot - 3 = 0$

Factor $x$ out of $x (6 x^{3} + x^{2}) + x (- 18 x)$ .

$x (6 x^{3} + x^{2} - 18 x) + x \cdot - 3 = 0$

Factor $x$ out of $x (6 x^{3} + x^{2} - 18 x) + x \cdot - 3$ .

$x (6 x^{3} + x^{2} - 18 x - 3) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$x ((6 x^{3} + x^{2}) - 18 x - 3) = 0$

Factor out the greatest common factor (GCF) from each group.

$x (x^{2} (6 x + 1) - 3 (6 x + 1)) = 0$

Factor.

Factor the polynomial by factoring out the greatest common factor, $6 x + 1$ .

$x ((6 x + 1) (x^{2} - 3)) = 0$

Remove unnecessary parentheses.

$x (6 x + 1) (x^{2} - 3) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$6 x + 1 = 0$

$x^{2} - 3 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $6 x + 1$ equal to $0$ and solve for $x$ .

Set $6 x + 1$ equal to $0$ .

$6 x + 1 = 0$

Solve $6 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$6 x = - 1$

Divide each term in $6 x = - 1$ by $6$ and simplify.

Divide each term in $6 x = - 1$ by $6$ .

$\frac{6 x}{6} = \frac{- 1}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 1}{6}$

Divide $x$ by $1$ .

$x = \frac{- 1}{6}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{6}$

Set $x^{2} - 3$ equal to $0$ and solve for $x$ .

Set $x^{2} - 3$ equal to $0$ .

$x^{2} - 3 = 0$

Solve $x^{2} - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$x^{2} = 3$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{3}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{3}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{3}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{3}, - \sqrt{3}$

The final solution is all the values that make $x (6 x + 1) (x^{2} - 3) = 0$ true.

$x = 0, - \frac{1}{6}, \sqrt{3}, - \sqrt{3}$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- \frac{1}{6}, 0), (\sqrt{3}, 0), (- \sqrt{3}, 0)$

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 6 {(0)}^{4} + {(0)}^{3} - 18 {(0)}^{2} - 3 \cdot 0$

Solve the equation.

Remove parentheses.

$y = 6 \cdot 0^{4} + {(0)}^{3} - 18 {(0)}^{2} - 3 \cdot 0$

Remove parentheses.

$y = 6 \cdot 0^{4} + 0^{3} - 18 {(0)}^{2} - 3 \cdot 0$

Remove parentheses.

$y = 6 \cdot 0^{4} + 0^{3} - 18 \cdot 0^{2} - 3 \cdot 0$

Remove parentheses.

$y = 6 {(0)}^{4} + {(0)}^{3} - 18 {(0)}^{2} - 3 \cdot 0$

Simplify $6 {(0)}^{4} + {(0)}^{3} - 18 {(0)}^{2} - 3 \cdot 0$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$y = 6 \cdot 0 + {(0)}^{3} - 18 {(0)}^{2} - 3 \cdot 0$

Multiply $6$ by $0$ .

$y = 0 + {(0)}^{3} - 18 {(0)}^{2} - 3 \cdot 0$

Raising $0$ to any positive power yields $0$ .

$y = 0 + 0 - 18 {(0)}^{2} - 3 \cdot 0$

Raising $0$ to any positive power yields $0$ .

$y = 0 + 0 - 18 \cdot 0 - 3 \cdot 0$

Multiply $- 18$ by $0$ .

$y = 0 + 0 + 0 - 3 \cdot 0$

Multiply $- 3$ by $0$ .

$y = 0 + 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0 + 0$

Add $0$ and $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

List the intersections.

x-intercept(s): $(0, 0), (- \frac{1}{6}, 0), (\sqrt{3}, 0), (- \sqrt{3}, 0)$

y-intercept(s): $(0, 0)$

Find the x and y Intercepts f(x)=6x^4+x^3-18x^2-3x

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